Magnetic susceptibility quantitation with MRI by solving boundary value problems

Magnetic susceptibility measurement is of considerable research interest in MRI and MRS. A rigorous method was previously developed to quantify the susceptibility of an arbitrarily shaped uniform object in an inhomogeneous external field. However, it requires using the field distribution information on a spherical surface or shell in the surrounding homogeneous medium enclosing the object. In this work, a new approach was developed through solving the boundary value problems of the Laplace equation, which has an advantage that the boundary providing the necessary field distribution information can have an arbitrary shape. This method has been validated on rectangular boundaries with both numerical simulation as well as experimental data. It has also been realized that MRI provides an experimental means of solving some boundary value problems of partial differential equations, if proper boundary condition can be set up.